Seems like the only distinction is that sinking fund is a broader term that also includes callable options?
Nope.
A sinking fund provision requires that some bonds be retired before maturity. For example, you may issue $10 million par of 10-year bonds with a sinking fund requirement that 20% of the bonds (or $2 million par) be retired every year starting in year 6. You may retire them by buying them on the open market, or you may retire them at par by random serial numbers. Apart from the sinking fund provision, the bonds are usually straight bonds; i.e., coupon only payments with par payable at maturity.
Amortizing bonds have part of the principal paid at each coupon date, so that the entire principal is paid as of the last coupon date (i.e., the issuer doesn’t have to make an additional principal payment at maturity). Partially amortizing bonds also have part of the principal paid at each coupon date, but there is still some principal to be paid at maturity.
Neither of these provisions has anything to do with the bonds being callable.
Ah, so it’s possible to have both amortizing/partially amortizing/bullet bonds and sinking fund bonds with call options, so why then make the distinction between a partially amortizing bond and a sinking fund? Why were these sinking fund bonds created? What is their advantage/disadvantage compared to amortizing/partially amortizing/bullet bonds?